Let us call the given central angle as theta.

Let us call the given radius of the circle as r.

Let call the upper "?" as x, and the lower "?" as y.

Let us call the hypotenuse of the legs x and y as z.

So the whole vertical line from the center of the circle up to the bottom of y is (r+z).

In the large right triangle, whose hypotenuse is (r+z):

cos(theta) = r / (r+z)

So,

(r+z)cos(theta) = r

(r+z) = r / cos(theta) = r*sec(theta)

z = r*sec(theta) -r

z = r*[sec(theta) -1] -------**

In the smaller right triangle, whose hypotenuse is z:

Since r and y are parallel ...they are both perpendicular to the same green line...then the angle between y and z is equal to theta.

Then,

sin(theta) = x / z

So,

x = z*sin(theta)

x = r[sec(theta) -1]*sin(theta) ------------answer.

cos(theta) = y / z

y = z*cos(theta)

y = r[sec(theta) -1]*cos(theta) --------answer.

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EDIT:

Or, x = r[1 /cos(theta) -1] sin(theta) = r[sin(theta)/cos(theta) -sin(theta)] = r[tan(theta) -sin(theta)]

And, y = r[cos(theta) /cos(theta) -cos(theta)] = r[1 -cos(theta)].